Yes, the center of a rigid spinning disk also spins. When a rigid object like a disk spins, all points on the object, including the center, undergo circular motion. Each point on the disk travels in a circular path around the axis of rotation, with the radius of the circular path increasing as you move farther from the axis.
To understand this, imagine drawing concentric circles on the disk, starting from the center and moving outward. As the disk spins, each circle rotates around the axis of rotation. Since the center of the disk is part of the disk itself, it will also rotate with the same angular velocity as the rest of the disk.
It's important to note that although all points on the disk rotate, the linear speed of each point differs. The linear speed is highest at the outer edge of the disk and decreases as you move towards the center. In contrast, the angular speed (measured in radians per second) remains constant throughout the disk, including at the center.